The value of `3xx 9^(-3//2) xx 9^(1//2)` is

Options

A. `(1)/(3)`
B. 3
C. 27
D. `-(1)/(3)`

Answer

To solve the expression \(3 \times 9^{-\frac{3}{2}} \times 9^{\frac{1}{2}}\), we can follow these steps:

### Step 1: Rewrite the base
First, we note that \(9\) can be expressed as \(3^2\). Therefore, we can rewrite the expression as:
\[ 3 \times (3^2)^{-\frac{3}{2}} \times (3^2)^{\frac{1}{2}} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we can simplify the exponents:
\[ 3 \times 3^{-3} \times 3^1 \]
This is because:
\[ (3^2)^{-\frac{3}{2}} = 3^{2 \cdot -\frac{3}{2}} = 3^{-3} \]
and
\[ (3^2)^{\frac{1}{2}} = 3^{2 \cdot \frac{1}{2}} = 3^1 \]

### Step 3: Combine the exponents
Now, we can combine the powers of \(3\) using the rule \(a^m \times a^n = a^{m+n}\):
\[ 3^{1} \times 3^{-3} \times 3^{1} = 3^{1 - 3 + 1} \]
Calculating the exponent:
\[ 1 - 3 + 1 = -1 \]
Thus, we have:
\[ 3^{-1} \]

### Step 4: Simplify the expression
The expression \(3^{-1}\) can be rewritten as:
\[ \frac{1}{3} \]

### Final Answer
Therefore, the value of \(3 \times 9^{-\frac{3}{2}} \times 9^{\frac{1}{2}}\) is:
\[ \frac{1}{3} \]